Face (geometry)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object;[1] a three-dimensional solid bounded exclusively by faces is a polyhedron.
A face can be finite like a polygon or circle, or infinite like a half-plane or plane.
[2] In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).
[3] In elementary geometry, a face is a polygon[note 1] on the boundary of a polyhedron.
[3][4] Other names for a polygonal face include polyhedron side and Euclidean plane tile.
Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope.
With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells.
This equation is known as Euler's polyhedron formula.
For example, a cube has 12 edges and 8 vertices, and hence 6 faces.
In higher-dimensional geometry, the faces of a polytope are features of all dimensions.
[3][5][6] A face of dimension k is called a k-face.
For example, the polygonal faces of an ordinary polyhedron are 2-faces.
For any n-polytope (n-dimensional polytope), −1 ≤ k ≤ n. For example, with this meaning, the faces of a cube comprise the cube itself (3-face), its (square) facets (2-faces), its (line segment) edges (1-faces), its (point) vertices (0-faces), and the empty set.
In some areas of mathematics, such as polyhedral combinatorics, a polytope is by definition convex.
Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P.[7] From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set.
[5][6] In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, the requirement for convexity is relaxed.
Abstract theory still requires that the set of faces include the polytope itself and the empty set.
An n-dimensional simplex (line segment (n = 1), triangle (n = 2), tetrahedron (n = 3), etc.
The number of them that are k-faces, for k ∈ {−1, 0, ..., n}, is the binomial coefficient
There are specific names for k-faces depending on the value of k and, in some cases, how close k is to the dimensionality n of the polytope.
Vertex is the common name for a 0-face.
The use of face in a context where a specific k is meant for a k-face but is not explicitly specified is commonly a 2-face.
A cell is a polyhedral element (3-face) of a 4-dimensional polytope or 3-dimensional tessellation, or higher.
Examples: In higher-dimensional geometry, the facets (also called hyperfaces)[8] of a n-polytope are the (n − 1)-faces (faces of dimension one less than the polytope itself).
For example: In related terminology, the (n − 2)-faces of an n-polytope are called ridges (also subfacets).
[10] A ridge is seen as the boundary between exactly two facets of a polytope or honeycomb.
For example: The (n − 3)-faces of an n-polytope are called peaks.
A peak contains a rotational axis of facets and ridges in a regular polytope or honeycomb.
where a linear functional achieves its minimum on
to be convex (else the boundary of a disc is a face of the disc, as well as any subset of the boundary) or closed.
